Emergence of Non-Linear Gravity from Relational Retrocausal Quantum Mechanics and Non-Commutative Matrix Models

The fundamental incompatibility between the linear, unitary evolution of quantum mechanics and the non-linear, deterministic geometry of General Relativity (GR) suggests a failure in our foundational assumptions regarding continuous spacetime and fixed macroscopic histories. In this paper, we demonstrate that the non-linearity of the Einstein Field Equations (EFE) and the apparent non-unitary collapse of the quantum wavefunction are dual manifestations of a single underlying principle: the preservation of relational algebraic rigidity across scales. We model spacetime usinga zero-dimensional Yang-Mills matrix framework where coordinates are non-commuting operators ([Xµ, Xν] = iθµν). We explicitly show how the failure of ultraviolet (UV) degrees of freedom todecouple—known as UV/IR mixing—dynamically forces the infrared (IR) effective action to adopt the non-linear curvature of the Einstein-Hilbert action. Extending this to Lorentzian signature introduces temporal non-commutativity ([X0, Xi] ̸= 0), which smears the effective propagator and inevitably violates strict time-ordering at the microscopic level. This non-local temporal smearing provides the exact mathematical foundation for Relational Retrocausal Quantum Mechanics(RRQM). Within RRQM, measurement is not a dynamical collapse, but a retroactive boundarycondition imposed upon the path integral, pruning the accessible history of the observer into global consistency. By synthesizing non-commutative geometry with RRQM’s Rewriting Postulate, we show that unitarity is equivalent to diffeomorphism invariance, successfully resolving the Wigner’sFriend paradox, Bell non-locality, and macroscopic superpositions without invoking many-worlds orhidden variables.